Home > Math > Background for My Critique of “Imagining the Tenth Dimension”

Background for My Critique of “Imagining the Tenth Dimension”

In my critique of Imagining the Tenth Dimension, I referenced a couple of concepts that you may not understand very well: The Mandelbrot Set and Aleph One. This is an attempt at explaining what you need to know about those concepts in order to understand what I was talking about.

First is the Mandelbrot set. It was proposed by Beniot Mandelbrot, who effectively invented fractal geometry and is one of the most brilliant mathematicians alive today. The Mandelbrot set is a type of fractal. What is a fractal? Well, a fractal is a type of object that has self-similarity. If you zoom in on the object, it looks similar to the object as a whole. This is very useful for modeling real-life systems. Look at rocks, for example. Have you ever noticed that a small rock looks a lot like a boulder? That, if you look up close at one part of a boulder, it looks like it itself could be a boulder? Trees exhibit the same behavior: a twig with leaves on it looks like a smaller version of a tree. This is true for many types of objects.

As explained by a song:

Take a point called z on the complex plane. Let z1 be z squared plus c. Let z2 be z1 squared plus c. Let z3 be z2 squared plus c. And so on. If the series of z’s will always stay close to z and never trend away, that point is in the Mandelbrot set.

So when you do this, you end up with a graph that looks rather complicated, but exhibits self-similar properties: if you zoom in on some spots, it looks the same as the whole picture.

How does this relate to dimensions? Well, traditionally the Mandelbrot set appears on a two-dimensional graph. There is one axis for the real numbers and one for the imaginary numbers. But the set is actually three-dimensional; the third dimension is simply not represented using a spatial dimension. Instead, it is represented using color.

What is this third dimension? Well, some areas of the graph are black. A black point is part of the Mandelbrot set. But if the point is not part of the Mandelbrot set, then it is given a color, where this color represents how long it takes for the point to diverge. This is a pretty useful example of representing a dimension using something other than space.

Aleph One

This concept takes much more explanation, so try to bear with me.

Infinity was not a particularly well-understood concept until the invention of Set Theory by Georg Cantor, around the end of the 19th century. Cantor was considered to be insane by many of his contemporaries, but that’s not really the point. The point is that he effectively defined how to look at different infinities.

Cantor defined infinity in terms of infinite sets. For any finite set with N elements, the infinite set has more elements than that. The simplest example here is the set of all natural numbers. This set contains the number 1, the number 2, the numbers 3, 4, 5, 6, and so on. Any natural number you can possibly think of is contained within this set.

There are also other infinite sets of the same size. (When two sets have the same number of elements, we say that they have the same cardinality.) To prove that two sets have the same cardinality, see if you can match all their elements in a one-to-one correspondence. If you can match every element in set A to an element in set B, then the sets have the same cardinality. This can be used to prove that two infinite sets are the same size.

From this we can infer that any set of numbers that can be listed out (even if it takes forever to finish) has a one-to-one correspondence with the natural numbers; that is, the sets have the same cardinality.

Look at the set of all integers (both positive and negative whole numbers). You could try listing 1, 2, 3, etc, and when you’re done you can list 0, -1, -2, -3, etc. But the problem is, you’re never done so you’ll never get to list the negative numbers.

But there is a solution. You can list out every integer by alternating: 0, 1, -1, 2, -2, 3, -3, etc. If you continue this process, it’s impossible to name an integer that won’t be contained within this set.

It’s also possible to list every fraction and mixed number; the actual process is rather difficult to explain in a blog format, so see this page for a fun explanation. (Edit: It looks like that site is down. See this one instead.)

What about the real numbers? Surely there must be some way to list them out. To make it simple, let’s just list every number from 0 to 1. Here is such a list, in no particular order.


0.018374020....
0.487384738....
0.297998172....
0.693729348....
0.382749207....
....

Since we are talking about all the real numbers (both rational and irrational), most of these numbers will have an infinite decimal expansion.

Now we can prove that there are more real numbers than naturals by showing that it is impossible to list every real number, even given an infinite amount of time. Move down the list in a diagonal path and change every digit to something else. For example, in the first number change that 0 to a 1, in the second number change the 8 to a 9, then change 7 to 8, 7 to 8, and 4 to 5. Use these changed digits to create a new number: 0.19885…. Once you go through the entire list, you’ll have a new number. Since this number is different from every other number by at least one digit, you now have a number that you did not include on your list. But your list was infinitely long; how is this possible?

What this means is that it is actually impossible to list out every real number. Therefore, there are actually more real numbers than there are natural numbers.

Sets of infinity are named using Aleph numbers. The first infinity, that of the natural numbers, is called aleph naught. The second infinity, the infinity of the reals, is called aleph one.

This should give you a good enough understanding of the concepts to understand their relevance to my original post. If I didn’t explain it well enough, or if you want to know more, you can find plenty of information on the World Wide Web.

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Categories: Math
  1. Dave
    June 12, 2010 at 3:08 pm

    Thanks, Michael. This explanation was concise and clear. Two of my favorite qualities in a verbal explanation of a mathematical concept.

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